Do You Need a GameCube Memory Card for Game Boy Games? No.

No — completely different systems.

• Game Boy / Color / Advance: Handhelds. Saves stored on the cartridge (battery-backed SRAM or Flash). No memory card slot.
• GameCube: Home console. Uses Memory Cards (59/251/1019 blocks) in slots A/B. For GameCube games only.
• Game Boy Player (GC accessory): Plays GBA carts on TV via GameCube. Saves still go to the GBA cartridge — not GC Memory Card.

If you're playing Game Boy games on a Game Boy, you need nothing extra. If on Game Boy Player, you need the GBA cart (with working battery). GC Memory Card is irrelevant.